Huber-Wasserstein barycenters for robust distribution-valued data

2026-08-13

AI Summary

This work addresses the sensitivity of the classical Wasserstein barycenter to outliers and its insufficient robustness. The authors propose a displacement-level Huberization approach that incorporates the Huber loss into the optimal transport cost, yielding a robust barycenter that exhibits locally quadratic behavior while suppressing large displacements. This formulation naturally interpolates between the Wasserstein mean and median and achieves a breakdown point as high as 1/2. Theoretical analysis establishes the existence, stability, and asymptotic distribution of the proposed estimator. Numerical experiments corroborate its strong robustness against contaminated data and confirm its interpolation properties between mean- and median-type aggregation.

Abstract

We propose a robust barycenter for distribution-valued data by incorporating the Huber loss directly into the optimal transport cost. In contrast to metric-space Huber means, which apply the Huber loss to the Wasserstein distance after optimization, our construction acts on individual transport displacements, preserving quadratic behavior locally while limiting the influence of large displacements. The resulting Huber-Wasserstein barycenters form a natural interpolation between Wasserstein means and $L^1$-type Wasserstein medians. We establish the analytical and statistical foundations of this construction. For optimal transport with Huber loss, we prove regularity and uniqueness properties of dual potentials, existence of optimal transport maps, and stability as the Huber parameter varies. For the associated barycenter problem, we prove existence and characterization results, consistency of empirical plug-in estimators, and a finite-sample breakdown point essentially equal to $1/2$. In dimension one, we further derive the pointwise influence function and asymptotic distribution, quantify the associated robustness-efficiency trade-off, and show that displacement-wise Huberization can retain first-order information that is lost by distance-based Huberization under localized shape contamination. Numerical experiments on contaminated distribution-valued data demonstrate the robustness of the proposed barycenters and illustrate their interpolation between mean- and median-like behavior.

Problem

• robust barycenter
• distribution-valued data
• Huber loss
• Wasserstein distance
• outlier contamination

Innovation

• Huber-Wasserstein barycenter
• robust optimal transport
• displacement-wise Huberization
• distribution-valued data
• breakdown point